C*-isomorphisms, Jordan isomorphisms, and numerical range preserving maps

Hwa Long Gau, Chi Kwong Li

Research output: Contribution to journalArticlepeer-review

28 Scopus citations


Let V = B(H) or S(H), where B(H) is the algebra of a bounded linear operator acting on the Hilbert space H, and S(H) is the set of selfadjoint operators in B(H). Denote the numerical range of A ∈ B(H) by W(A) = {(Ax,x): x ∈ H, (x,x) = 1}. It is shown that a surjective map φ: V → V satisfies W(AB + BA) = W(φ(A)φ(B) + φ(B)φ(A)) for all A, B ∈ V if and only if there is a unitary operator U ∈ B(H) such that φ has the form X ±U*XU or X ±U*Xt U, where Xt is the transpose of X with respect to a fixed orthonormal basis. In other words, the map φ or-φ is a C*-isomorphism on B(H) and a Jordan isomorphism on S(H). Moreover, if H has finite dimension, then the surjective assumption on φ can be removed.

Original languageEnglish
Pages (from-to)2907-2914
Number of pages8
JournalProceedings of the American Mathematical Society
Issue number9
StatePublished - Sep 2007


  • Jordan product
  • Numerical range


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